A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Monotone Class Theorem Application

I am trying to proof the following statement. Let $h$ be a bounded, $\mathbb{F}$-predictable process with $\tau$ a $\mathbb{H}$-stopping time, we then like to prove \begin{equation} ...
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8 views

Bivariate GBM - crosscovariance

I have troubles concerning a correlated bivariate GBM with identical drift and diffusion rates. Let $dX^i_t = \mu X^i_t dt + \sigma X^i_tdW^i_t$ and $E[dW_t ^idW^j_t] = \rho_{i,j}dt$ If $S_0^i = ...
2
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2answers
19 views

Lévy's upward theorem and $\mathcal{L}^p$ convergence.

Lévy's upward theorem: Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} ...
2
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0answers
10 views

Trying to show convergence (in probability) of integrals using taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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10 views

How close is a Ornstein-Uhlenbeckprocess to Brownian Motion

The Semi-Variance function of an Ornstein-Uhlenbeck (OU) process can be written as: $\gamma(\tau) = \sigma * (1 - \exp(\frac{-\tau}{a})$. If $a \to \infty$ the OU-Process approaches Brownian motion ...
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13 views

Self similar process

I am learning long memory process and came cross the definition of self similar. By definition, process $X(t)$ is self similar if $X(at)=_d a^H X(t)$,$a>0$ and $H$ is Hurst exponent. By equality of ...
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1answer
14 views

Is it an increasing process?

On a probability space $(\Omega,\mathscr{F},\mathbb{P})$ with filtration generated by Brownian motion, there is a progressivley process $(A_t)_{t\in[0,T]}$. If for any stopping times $0\leq \sigma\leq ...
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22 views

Why Gaussian process is not Ergodic in general?

Can anyone use a simple way to explain this? I heard this in class but I do not know why. By Wiki: a random process is ergodic if its statistical properties can be deduced from a ...
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13 views

How can I know by inspection that a process is WSS?

I have some codes to generate three different Random Sequences: I am getting a 4x100 matrixes where 4 is the number of samples and 100 is the length of the process. I am getting these results: ...
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7 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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1answer
24 views

Marginal Probability of Stochastic Process

I have a wide sense stationary stochastic process x(t)=asin(2πf0t)+bcos(2πf0t) where a & b are independent gaussian random variables. How can I find the Marginal probability of x(t)? I am ...
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1answer
11 views

Asymmetric simple random walk?

It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of $(a,b)$? Can anyone help? Thanks so much!
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1answer
21 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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1answer
26 views

Property of submartingale and supermartingle?

Is it true that for a submartingale, $$E(X_n) \le E(X_m)$$ for $n \le m$. And for a supermartingale, $$E(X_n) \ge E(X_m)$$ for $n \le m$. If it is true, then why? I feel confused because the ...
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1answer
19 views

Markov property of Brownian motion

There are two statements about Markov property: $B_t $ is Brownian motion and $\mathcal{F}$ is generated by $B$ If $s>0$ and $Y$ is bounded and measuable, then ...
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33 views

Where is the assumption of right continuity used in the following proof?

Lemma:If $X$ be a right-continuous positive local martingale then , $X$ is a generalized super martingale Proof: $\forall s<t$ $$E[X_t\mid F_s]=E[\lim_{n\to\infty} X_{t \wedge\tau_n}\mid F_s] \leq ...
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23 views

Birth immigration process

I'm having some problem with this question. A model for the distribution of the number of goals scored in soccer matches suggests that if n goals have already been scored by time t, then the ...
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15 views

If a stochastic process follows Geometric Brownian Motion, does it imply that it is Log-normally distributed and vice-versa?

This might be a naive question, but it doesn't stop haunting me. Wiki page for GBM writes the SDE for GBM process and shows it follows log-normal distribution. Is it true every time or are there any ...
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1answer
24 views

A die whose score cannot be as before (Markov chains)

A die is "fixed" so that each time it is rolled the score cannot be the same as the preceding score, all other scores having probability $1/5$. Given that the first score is 6, what is the probability ...
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18 views

Translated stochastic process

Let $M$ be a (compact) Riemannian manifold and let $L$ be some second-order elliptic operator on $M$. Now for a vector field $v$, I can consider the flow $\Psi_t$ of $v$ and consider the following ...
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29 views
+150

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
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0answers
16 views

Proving convergence of a martingale in $L^2$ [on hold]

I'm stuck with the following problem: Let $X$ a positive martingale bounded in $L^2$. Show that $\lim_{n\to \infty} X_n = X$ a.s. and in $L^2$.
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1answer
31 views

Cellular automata (Random walk)

Here is the context of my question below. I cite from "Some Rigorous Results for the Greenberg-Hastings Model" by Richard Durrett- Consider the following cellular automaton known as the ...
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22 views

show is markov chain [on hold]

suppose: that : X=({X}{n}){n\geq 0}: is: M.C(\lambda ,P): y : f:IxI\rightarrow I a function. denote by ${f}^{-1}(j):={i\in I:f(i)=j}\: \: y \:$ suppose for all $i,j \in I$ such that $ f(i)=f(j)$ ...
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49 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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0answers
26 views

Limit of stochastic process [on hold]

trying to get my hand on the limit of these probabilities edit : first one how do i deal with all that inf
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1answer
23 views

Marginal probability density function of Stochastic process

I was solving the following question and I derived the Auto correlation function and proved that it is a WSS process. However, I am not sure how to go about finding the Marginal probability density ...
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0answers
13 views

Finding the infinitesimal generator of a M/M/2 queue [on hold]

I have a M/M/2 queue with a total population of 5. The arrival times are independent exponential random variables with mean of $\lambda$ and the service times have a mean of $\mu$. The initial number ...
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6 views

Stochastic Process with mean reverting property

Here I am seeking for a definition of what kind of stochastic processes are called mean reverting stochastic process. That is, what are the properties that a stochastic process should obey in order to ...
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20 views

Usual augmentation filtration? (Sigma algebra generated by a descreasing family of sets)?

My aime is to understand the usual augmentation filtration. More pricesely, I want to understand the last identity in this PDF file. http://onlinelibrary.wiley.com/doi/10.1002/0470863617.app1/pdf ...
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41 views

probability in process [on hold]

We have a series of $N$ cars . Cars lengths are distributed like the time between following events of poisson process $(X_t)t>0$ with rate $p$ . We sample the length of a car far ...
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1answer
15 views

Finding mean and variance of stochastic process

If I'm given a Stochastic Process Xt that satisfies a stochastic diff. equation, let's say fXt, what is the formula to find the mean and variance of Xt? I think it's: $mean = dE(X_t) = dX_0e^t$ ...
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1answer
27 views

Algebra and partions of a set

My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every ...
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29 views

Martingales relative to its natural filtration [on hold]

Let {$Y_n$} be a sequence of positive independent random variables with E($Y_j$) =1 for all j. Set $X_0$=1 and $X_n = \prod_{j=1}^n Y_j$, $n \geq 1$. How can I show $X_0,X_1,X_2,...$ is a martingale ...
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17 views

little's law process [on hold]

little's law says that L = GW where G is arrival times average rate W is average staying time in the system and L is the average number of people in the systerm . if we take a look at the process ...
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0answers
13 views

Expected Value of a random process [on hold]

I can't for the life of me figure this out. Consider the random process x(t)=1/2 +(1/2)$\cos$(wt+$\theta$), where $\theta$ is uniform on the interval [0,2$\pi$] Calculate Expected value Calculate ...
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2answers
31 views

quadratic variations of Brownian motion squared

I'm trying to refresh my memories about stochastic processes. We know that Brownian motion has as quadratic variation equals to t. What is the quadratic variation of the Brownian motion squared ? ...
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1answer
28 views

I want to find Sigma-field generated by $X=5I_{\{1,3,7\}}+6I_{\{4,5\}}+7I_{\{8,9\}}$. with follow condition. [closed]

Let $\Omega=\{1,\dots,10\}$ and $\mathcal F= 2^\Omega$ and we have $$P(\{1\})=P(\{3\})=P(\{7\})=\frac{1}{15}$$ $$P(\{2\})=P(\{6\})=P(\{10\})=\frac{1}{6}$$ $$P(\{4\})=P(\{5\})=\frac{1}{20}$$ ...
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25 views

Monte Carlo Markov Chain simulation

I am going to post the python code logic we used however I want someone to look at the number that are printing out. The Markov chain is uniformly distributed across all $50x50$ matrices with entries ...
3
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1answer
48 views

Recurrence/Transience of random walk with +2/-1 steps

Consider the Markov chain with state space $S=(0,1,2,...)$ and transition probabilities: $p(x,x+2)=p$ , $p(x,x-1)=1-p$, $\forall$ $x>0$. $p(0,2)=p$ , $p(0,0)=1-p$. For which values of $p$ is this ...
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3answers
37 views

an exercise about mean and probability

Let $(\Omega, \mathcal{F}, P)$ be a probability space, $X : \Omega \rightarrow \mathbb{R}$ be a discrete random variable and $$\phi : [0, \infty) \rightarrow (0, \infty)$$ an increasing function (so ...
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1answer
35 views

$E[1_{\lbrace P_T-P_{\tau_n}=0\rbrace}\int_{\tau_n}^T h(s)dN_s]=0?$

If $P_t$ is a standard Poisson process, and $N_t=P_t-t$ the associated martingale then $\int_0^t h(s)dN_s$ is a martingale (assuming that h satisfies the neccessary hypothesis). Thus, considering ...
3
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1answer
40 views

Inequality for the expected values of norm of stochastic processes

Let $\underline{X}=(x_1, x_2, x_3), \; x_i \sim \mathcal{N(0,1)}$ i.i.d. For any fixed $t>0$ and $\underline{X}_0$ prove that the following holds ($\Vert\cdot\Vert$ is the Euclidean norm): ...
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29 views

Optional Sampling Theorem Application

Let x, y > 0. Define the first passage time of a Brownian motion $W_t$ as $\tau_a$ = min{t $\ge$ 0: $W_t$ = a}. I need to show that $$E[e^{-u\tau_x}1_{\tau_x < \tau_{-y}}] = ...
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0answers
15 views

equality involving the arrival times of a poisson process

Let $P_t$ be a Poisson process with arrival times $\tau_1,\tau_2,\dots$ and $h$ a bounded function, $F$ a square integrable function of the arrival times of $P_t$ until the time T . I am wondering if ...
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0answers
20 views

Can we have cases where the $P(x,y)$ does not exist? [closed]

I had encountered such assumption: Assume there exists a probability distribution $P(x,y)$ on $X \times Y$ which governs the data generation and underlying functional dependencies. My question is: ...
2
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1answer
25 views

distribution of $\sup\limits_{0\le t \le 1}|W(t)|$

My prof on class told us that distribution of $S=\sup\limits_{0\le t \le 1}|W(t)|$ has been well studied, where $W$ is a Wiener process, but I need a table to find $c$ such that $P(S < c) = 0.95$. ...
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30 views

Bivariate stopped processes

Take two dependent Levy processes $L_1(t)$ and $L_2(t)$ with law $\mathcal{L}(L_1(1),L_2(1)$. If we stop the first process at a general time $t=s_1$ and stop the second process at another general time ...
2
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1answer
23 views

Showing that if $B_t$ is a Brownian motion then $t B_{1/t}$ is Gaussian

I want to show that if $B_t$ is a Brownian motion then $t B_{1/t}$ is a Gaussian process, i.e. that it has increments which have the normal distribution. It seems like a trivial fact, since the ...
1
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1answer
34 views

Stochastic differential of Bessel process [closed]

Let $ \underline{B}_{t}=(B_1(t), \dots, B_d(t))$ be a $d$-dimensional Brownian motion. How to calculate the stochastic differential of $ \Vert{\underline{B}_t}\Vert$? $\Vert . \Vert$ denotes the ...